Short answer. Yes, generally — and the charter doesn't need to specifically allow it. Article 1026 of the Civil Code provides that private corporations for religious, scientific, cultural, educational, or charitable purposes may receive testamentary dispositions, and that all other corporations may succeed under a will unless their charter or the law of their creation says otherwise.
What the law says
A testamentary disposition may be made to the State, provinces, municipal corporations, private corporations, organizations, or associations for religious, scientific, cultural, educational, or charitable purposes
Civil Code, Article 1026 — Dispositions to Institutions. Read the full provision →
What the law says
All other corporations or entities may succeed under a will, unless there is a provision to the contrary in their charter or the laws of their creation, and always subject to the same
Civil Code, Article 1026 — Dispositions to Institutions. Read the full provision →
Corporations for religious, scientific, cultural, educational, or charitable purposes
Article 1026 of the Civil Code opens by naming a specific group of eligible recipients: "A testamentary disposition may be made to the State, provinces, municipal corporations, private corporations, organizations, or associations for religious, scientific, cultural, educational, or charitable purposes." A private corporation organized for one of those listed purposes is expressly named as someone a testator may leave property to.
All other corporations: allowed unless their own charter says no
The article's second sentence covers everything else: "All other corporations or entities may succeed under a will, unless there is a provision to the contrary in their charter or the laws of their creation, and always subject to the same." The structure runs the opposite direction from what your question assumes — the default is that a corporation may succeed, and it takes an actual contrary provision in its own charter or creating law to take that capacity away.
Why this matters for an ordinary private company
An ordinary private company, not organized for a religious, scientific, cultural, educational, or charitable purpose, falls into this second, broader category rather than the first. That does not disqualify it — the second sentence gives it the same general capacity to succeed under a will as any other corporation, subject only to whatever its own charter or the law of its creation actually says. Nothing in Article 1026 requires that capacity to be spelled out affirmatively in the charter before it exists.
What to actually check for a specific company
Because the article defers to "their charter or the laws of their creation," the real answer for any particular company depends on documents this article does not itself contain — its articles of incorporation, by-laws, or the special law that created it, if it was created by one. If none of those actually contain a provision against succeeding under a will, Article 1026 leaves the company's capacity to be named an heir intact by default.